Positive solutions for second-order differential equations of Kirchhoff type on the half-line


The aim of the present paper is to study the existence of nontrivial nonnegative solutions for asecond-order boundary value problem of Kirchhoff type on the half-line. Our approach is based on variational methods, a monotonicity trick related to the mountain pass lemma, cut-off functional technique, and a Pohozaev type identity.


Habiba Boulaiki
Faculty of Mathematics USTB, El-Alia Bar-Ezzouar Algiers, Algeria

Toufik Moussaoui
Ecole Normale Superieure, Kouba, Algiers, Algeria

Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania



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Habiba Boulaiki  Toufik Moussaoui, R. Precup, Positive solutions for second-order differential equations of Kirchhoff type on the half-line, Carpathian J. Math. 37 (2021), No. 2., 325-338, http://dx.doi.org/10.37193/CJM.2021.02.17


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Carpathian J. Math.

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1584 – 2851

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1843 – 4401

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[1] Bonanno, G. and O’Regan, D., A boundary value problem on the half-line via critical point methods, Dynam. Systems Appl., 15 (2006), 395–408
[2] Boulaiki, H., Moussaoui, T. and Precup, R., Multiple positive solutions for a second-order boundary value problem on the half-line, J. Nonlinear Funct. Anal., 2017 (2017), 1–25
[3] Brezis, H., Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2010
[4] Caristi, G., Heidarkhani, S. and Salari, A., Variational approaches to Kirchhoff-type second-order impulsive differential equations on the half-line, Results Math., 73 (2018), No. 1, Paper No. 44, 31 pp.
[5] Dickey, R. W., The initial value problem for a non linear semi-infinite string, Proc. Roy. Soc. Edinburgh Sect. A, 82 (1978), 19–26
[6] Duan, Y. and Zhou, Y., Existence of solutions for Kirchhoff type equations with unbounded potential, Electron. J. Differential Equations, 2017 (2017), 184, 1–12
[7] Gomes, J. M. and Sanchez, L., A variational approach to some boundary value problems in the half-line, Z. Angew. Math. Phys., 56 (2005), 192–209
[8] Greenberg, J. M. and Hu, S. C., The initial-value problem for a stretched string, Quart. Appl. Math., 38 (1980), 289–311
[9] Heidarkhani, S., Afrouzi, G. A. and Moradi, S., Existence results for a Kirchhoff type second-order differential equation on the half-line with impulses, Asymptot. Anal., 105 (2017), 137–158
[10] Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on RN , Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787–809
[11] Jeanjean, L., Local condition insuring bifurcation from the continuous spectrum, Math. Z., 232 (1999), 651–664
[12] Jeanjean, L. and Le Coz, S., An existence stability result for standing waves of nonlinear Schrodinger equations, Adv. Differential Equations,11 (2006), 813–840
[13] Kikuchi, H., Existence and stability of standing saves for Schrodinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403–437
[14] Kirchhoff, G., Vorlesungen Uber Mechanik, Teubner, Leipzig, 1883
[15] Li, Y., Li, F. and Shi, J., Existence of positive solutions to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285–2294
[16] Li, Y., Li, F. and Shi, J., Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl., 410 (2014), 361–374
[17] Ma, R. and B. Zhu, B., Existence of positive solutions for a semipositone boundary value problem on the half-line, Comput. Math. Appl., 58 (2009), 1672–1686
[18] Perera, A. K. and Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246–255
[19] Struwe, M., Variational Methods, Springer, New York, 1996
[20] Zima, M., On positive solution of boundary value problems on the half-line, J. Math. Anal. Appl., 259 (2001), 127–136


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